Indian National Mathematical Olympiad is organized by Homi Bhabha Centre for Science Education. This post is dedicated for INMO 2019 Discussion. You can post your ideas here.

Problem 1. Let $$ABC$$ be a triangle with $$\angle BAC > 90^\circ$$. Let $$D$$ be a point on the line segment $$BC$$ and $$E$$ be a point on the line $$AD$$ such that $$AB$$ is tangent to the circumcircle of triangle $$ACD$$ at $$A$$ and $$BE$$ perpendicular to $$AD$$. Given that $$CA=CD$$ and $$AE=CE$$, determine $$\angle BCA$$ in degrees

Problem 2. Let $${A_1}{B_1}{C_1}{D_1}{E_1}$$ be a regular pentagon. For $$2\leq n \leq 11$$, let $${A_n}{B_n}{C_n}{D_n}{E_n}$$ whose vertices are the midpoints of the sides of $${A_{n-1}}{B_{n-1} }{C_{n-1} }{D_{n-1}}{E_{n-1}}$$. All the $$5$$ vertices of each of the $$11$$ pentagons are arbitrarily coloured red or blue. Prove that four points among these $$55$$ have the same colour and form the vertices of a cyclic quadrilateral.

Problem 3. Let $$m$$, $$n$$ be distinct positive integers. Prove that $$\gcd (m,n) + \gcd (m+1,n+1) + \gcd (m+2,n+2) \leq 2|m-n|+1$$. Further, determine when equality holds.

Problem 4. Let $$n$$ and $$M$$ be positive integers such that $$M>n^{n-1}$$. Prove that there are $$n$$ distinct primes $$p_1$$,,,….,$$p_n$$ such that $$p_j$$ divides $$M+j$$ for $$1 \leq j \leq n$$.

Problem 5. Let $$AB$$ be a diameter of a circle $$\Gamma$$ and let $$C$$ be a point on $$\Gamma$$ different from $$A$$ and $$B$$. let $$D$$ be the foot of perpendicular from $$C$$ on to $$AB$$. let $$K$$ be a point of the segment $$CD$$ such that $$AC$$ is equal to the semiperimeter of the triangle $$ADK$$. Show that the excircle of triangle $$ADK$$ opposite $$A$$ is tangent to $$\Gamma$$.

Problem 6. Let $$f$$ be a function defined from the set $$\{(x,y) : x,y$$ are real, $$xy \neq 0\}$$ to the set of all positive real number such that

(i)$$f(xy,z)=f(x,z)f(y,z),$$ for all $$x,y \neq 0$$

(ii) $$f(x,1-x)=1,$$ for all $$x \neq 0,1$$

Prove that

(i)$$f(x,x)=f(x,-x)=1,$$ for all $$x \neq 0$$

(ii) $$f(x,y)f(y,x)=1,$$ for all $$x,y \neq 0$$